Skip to main content
SpringerLink
Log in

Algebras with involution that become hyperbolic over the function field of a conic

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q,−), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Kr. Arason, Cohomologische Invarianten quadratischer Formen, Journal of Algebra 36 (1975), 448–491.

    Article  MATH  MathSciNet  Google Scholar 

  2. E. Bayer-Fluckiger, D. Shapiro and J.-P. Tignol, Hyperbolic involutions, Mathematische Zeitschrift 214 (1993), 461–476.

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Bayer-Fluckiger, R. Parimala and A. Quéguiner-Mathieu, Pfister involutions, Indian Academy of Sciences. Proceedings. Mathematical Sciences 113 (2003), 365–377.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. J. Becher, A proof of the Pfister Factor Conjecture, Inventiones Mathematicae 173 (2008), 1–6.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Dejaiffe, Formes antihermitiennes devenant hyperboliques sur un corps de déploiement, Comptes Rendus Mathématique Série I 332 (2001), 105–108.

    MATH  MathSciNet  Google Scholar 

  6. I. Dejaiffe, Somme orthogonale d’algèbres à involution et algèbre de Clifford, Communications in Algebra 26 (1995), 1589–1612.

    Article  MathSciNet  Google Scholar 

  7. R. S. Garibaldi, Clifford algebras of hyperbolic involutions, Mathematische Zeitschrift 236 (2001), 321–349.

    Google Scholar 

  8. R. S. Garibaldi, Orthogonal involutions on algebras of degree 16 and the Killing form of E 8 (with an appendix by K. Zainoulline), in Quadratic Forms-Algebra, Arithmetic, and Geometry, Contemporary Mathematics, Vol. 493, Amererican Mathematical Soceity, Providence, RI, 2009, pp. 131–162.

    Google Scholar 

  9. P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, Cambridge, 2006.

    Book  MATH  Google Scholar 

  10. D. W. Hoffmann, Function fields of quadratic forms, Doctoral dissertation, University of California, Berkeley, California, 1992.

    Google Scholar 

  11. D. W. Hoffmann, D.W. Lewis and J. Van Geel, Minimal forms for function fields of conics, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992), Proceedings of Symposia in Pure Mathematics, Vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 227–237.

    Google Scholar 

  12. D. W. Hoffmann and J. Van Geel, Minimal forms with respect to function fields of conics, Manuscripta Mathematica 86 (1995), 23–48.

    Article  MATH  MathSciNet  Google Scholar 

  13. M.-A. Knus, T. Y. Lam, D. Shapiro and J.-P. Tignol, Discriminants of involutions on biquaternion algebras, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992), Proceedings of Symposia in Pure Mathematics, Vol. 58, Part 2, Amer. Math. Soc., Providence, RI, 1995, pp. 279–303.

    Google Scholar 

  14. M.-A. Knus, A. S. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, American Mathematical Society Colloquium Publications, Vol. 44, American Mathematical Society, Providence, RI, 1998.

    MATH  Google Scholar 

  15. T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society, Providence, RI, 2005.

    MATH  Google Scholar 

  16. A. S. Merkurjev, Simple algebras and quadratic forms, Izv. Akad. Nauk SSSR, Ser. Mat. 55 (1992), no. 1, 218–224; English translation: Math. USSR-Izv. 38 (1992), no. 1, 215–221.

    MathSciNet  Google Scholar 

  17. R. Parimala, R. Sridharan and V. Suresh, Hermitian analogue of a theorem of Springer, Journal of Algebra 243 (2001), 780–789.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. S. Pierce, Associative Algebras, Springer, New York, 1982.

    MATH  Google Scholar 

  19. W. Scharlau, Quadratic and Hermitian Forms, Springer, Berlin, 1985.

    MATH  Google Scholar 

  20. A. S. Sivatski, Applications of Clifford algebras to involutions and quadratic forms, Communications in Algebra 33 (2005), 937–951.

    Article  MATH  MathSciNet  Google Scholar 

  21. D. Tao, The generalized even Clifford algebra, Journal of Algebra 172 (1995), 184–204.

    Article  MATH  MathSciNet  Google Scholar 

  22. J.-P. Tignol, Réduction de l’indice d’une algèbre simple centrale sur le corps des fonctions d’une quadrique, Bulletin de la Société Mathématique de Belgique. Série A 42 (1990), 735–745.

    MATH  MathSciNet  Google Scholar 

  23. J.-P. Tignol, A Cassels-Pfister theorem for involutions on central simple algebras, Journal of Algebra 181 (1996), 857–875.

    Article  MATH  MathSciNet  Google Scholar 

  24. J.-P. Tignol and A. R. Wadsworth, Value functions and associated graded rings for semisimple algebras, Transactions of the American Mathematical Society 369 (2010), 687–726.

    MathSciNet  Google Scholar 

  25. J.-P. Tignol and A. R. Wadsworth, Valuations on algebras with involution, preprint arXiv:0902.0849.

Download references

Author information

Authors and Affiliations

  1. Université Paris 13 (LAGA), CNRS (UMR 7539), Université Paris 12 (IUFM), 93430, Villetaneuse, France

    Anne Quéguiner-Mathieu

  2. Département de Mathématiques, Université catholique de Louvain, Chemin du cyclotron, 2, B1348, Louvain-la-Neuve, Belgique

    Jean-Pierre Tignol

Authors
  1. Anne Quéguiner-Mathieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Pierre Tignol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anne Quéguiner-Mathieu.

Additional information

The first author thanks S. Garibaldi and Ph. Gille for motivating discussions and the Université Catholique de Louvain for its hospitality during the preparation of this paper. The second author gratefully acknowledges the hospitality of Université Paris 13, where part of the work leading to this paper was carried out. He was supported in part by the F.R.S.-FNRS. Both authors acknowledge support from Wallonie-Bruxelles International and the French goverment in the framework of “Partenariats Hubert Currien.” They thank the referee for his careful reading of the paper and valuable advice.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Quéguiner-Mathieu, A., Tignol, JP. Algebras with involution that become hyperbolic over the function field of a conic. Isr. J. Math. 180, 317–344 (2010). https://doi.org/10.1007/s11856-010-0106-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-010-0106-x

Keywords

Search

Navigation